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5.80 Small-Molecule Spectroscopy and Dynamics
Fall 2008
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Lecture # 22 Supplement
See Microwave Spectroscopy by C. H. Townes and A. L. Schawlow, Dover Publications, New York (1975)
for complete text of these Appendices.
Contents
A.
A.
Appendix III: Coefficients for Energy Levels of a Slightly Asymmetric Top, pp. 522­
526 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
B.
Appendix IV: Energy Levels of a Rigid Rotor, pp. 527-555 . . . . . . . . . . . . . . .
2
C.
Appendix V: Transition Strengths for Rotational Transitions, pp. 557-559 . . . . . .
2
Appendix III: Coefficients for Energy Levels of a Slightly Asym­
metric Top, pp. 522-526
SUMMARY
Rotational energy is given by
w = K 2 + C1 b + C2 b2 + C3 b3 + C4 b4 + C5 b5 + . . .
�
B + C�
B+C
J(J + 1) + A −
w
2
2
C−B
b = bp =
2A − B − C
�
A+B
A + B�
For an oblate top, energy = W =
J(J + 1) + C −
w
2
2
A−B
b = b0 =
2C − B − A
For a prolate top, energy = W =
Where the first few constants K, C1 , C2 . . . are identical for pairs of degenerate levels, they are usually
listed for only the first of the two levels. (C1 , C2 , and C3 were computed by J. F. Lotspeich; C4 and C5 by
J. Kraitchman and N. Solimene.)
1
Lecture #23 Supplement
Page 2
B. Appendix IV: Energy Levels of a Rigid Rotor, pp. 527-555
SUMMARY
Energy (in cycles/sec) = W/h =
1
(A
2
+ C)J(J + 1) + 12 (A − C)Eτ . Eτ is tabulated as a function of the
rotational level JK−1 K1 (or Jτ ) and of the asymmetry parameter κ =
2B−A−C
.
A−C
Values for positive κ only are tabulated, since those for negative κ can be obtained from the relation
Eτ (κ) = −E−τ (−κ). For further explanation see Chapter 4.
This table was reproduced with the permission of the Ballistics Research Laboratories, Aberdeen, MD,
from Ballistics Research Laboratories Report No. 878 (September, 1953), by T. E. Turner, B. L. Hicks,
and G. Reitwiesner. It was prepared for reproduction by S. Poley with the aid of an IBM card-controlled
typewriter at the Watson Scientific Computing Laboratory.
Tables of Eτ for J up to 40 and values of κ = 0, 0.1, 0.2, 0.3, . . . 1.0 are given by G. Erlandsson, Arkiv för
Fysik, to be published.
C. Appendix V: Transition Strengths for Rotational Transitions, pp.
557-559
SUMMARY
�
Intensity of a transition between rotational levels Jkl and Jmn
is proportional to
��
��2
� (κ) = (2J + 1) �(µ x ) J J � �
(µ x )2 x S Jkl Jmn
kl mn
Here µ x is the dipole moment along one of the principal axes of inertia (x = a, b or c), and S is the quantity
tabulated here as a function of initial and final state and of the asymmetry parameter κ. However, each
value has been multiplied by 104 to eliminate decimal points. The upper sign for values of K applies to
transition subbranches listed in the two left-hand columns, and the lower sign to those in the right-hand
columns. The axis along which a dipole moment is required to produce a given transition is indicated by
a superscript to the left of the subbranch designation. Thus c Q10 indicates a Q branch (ΔJ = 0) with a
change in K−1 of 1, a change in K1 of 0, and that a dipole moment µc along the c axis is required for the
transition. For further discussion see Chapter 4. [Tables in this appendix are taken from P. C. Cross, R. M.
Hainer, and G. W. King, J. Chem. Phys. 12, 210 (1944).]